Abstract

Studies of the electrooptic effect in potassium tantalate niobate (KTN) and Li doped KTN in the vicinity of the ferroelectric phase transition are reported. It was observed that in KTN the standard electrooptic behavior is accompanied by electrically induced depolarization of the light traversing through the crystal. This behavior is attributed to the influence of the fluctuating dipolar clusters that are formed in KTN above the ferroelectric phase transition due to the emergence of the Nb ions out of the center of inversion of the unit cell. It was shown in addition that this behavior is inhibited in Li doped KTN, which enables exploiting the large electrooptic effect in these crystals.

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References

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  1. S. E. Lerner, P. Ben Ishai, A. J. Agranat, and Yu. Feldman, “Percolation of polar nanoregions: a dynamic approach to the ferroelectric phase transition,” J. Non-Cryst. Solids 353(47-51), 4422–4427 (2007).
    [CrossRef]
  2. A. J. Agranat, M. Razvag, M. Balberg, and V. Leyva, “Dipolar holographic gratings induced by the photorefractive process in potassium lithium tantalate niobate at the paraelectric phase,” J. Opt. Soc. Am. B 14(8), 2043–2048 (1997).
    [CrossRef]
  3. A. J. Agranat, M. Razvag, M. Balberg, and G. Bitton, “Holographic gratings by spatial modulation of the Curie-Weiss temperature in photorefractive K1-xLixTa1-yNbyO3:Cu,V,” Phys. Rev. B 55(19), 12818–12821 (1997).
    [CrossRef]
  4. G. Bitton, M. Razvag, and A. J. Agranat, “Formation of metastable ferroelectric clusters in K1-xLixTa1-yNbyO3:Cu,V at the paraelectric phase,” Phys. Rev. B 58(9), 5282–5286 (1998).
    [CrossRef]
  5. E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83(10), 1954–1957 (1999).
    [CrossRef]
  6. E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nano-disordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011).
    [CrossRef]
  7. A. Bitman, N. Sapiens, L. Secundo, A. J. Agranat, G. Bartal, and M. Segev, “Electroholographic tunable volume grating in the g44 configuration,” Opt. Lett. 31(19), 2849–2851 (2006).
    [CrossRef] [PubMed]
  8. A. Yariv and P. Yeh, Optical Waves in Crystals (John Wiley & Sons, 1984), Chapter 7.3.1.
  9. R. Hofmeister, A. Yariv, and A. Agranat, “Growth and characterization of the perovskite K1-yLiyTa1-xNbxO3:Cu,” J. Cryst. Growth 131(3-4), 486–494 (1993).
    [CrossRef]
  10. M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon, 1977), Chapter 4.
  11. Y. Girshberg and Y. Yacoby, “Off-centre displacements and ferroelectric phase transition in dilute KTa1−xNbxO3,” J. Phys. Condens. Matter 13(39), 8817–8830 (2001).
    [CrossRef]
  12. G. Bitton, Yu. Feldman, and A. J. Agranat, “Relaxation processes of off-center impurities in KTN:Li crystals,” J. Non-Cryst. Solids 305(1-3), 362–367 (2002).
    [CrossRef]
  13. J. Toulouse, “The three characteristic temperatures of relaxor dynamics and their meaning,” Ferroelectrics 369(1), 203–213 (2008).
    [CrossRef]
  14. R. Blinc and B. Zeks, Soft Modes in Ferroelectrics and Antiferroelectrics (Elsevier, 1974).
  15. P. Ishai, C. de Oliveira, Y. Ryabov, Y. Feldman, and A. Agranat, “Glass forming liquid kinetics manifested in a KTN:Cu crystal,” Phys. Rev. B 70(13), 132104 (2004).
    [CrossRef]
  16. A. J. Agranat, “Optical lambda-switching at telecom wavelengths based on electroholography,” Top. Appl. Phys. 86, 133–161 (2003).
    [CrossRef]

2011 (1)

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nano-disordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011).
[CrossRef]

2008 (1)

J. Toulouse, “The three characteristic temperatures of relaxor dynamics and their meaning,” Ferroelectrics 369(1), 203–213 (2008).
[CrossRef]

2007 (1)

S. E. Lerner, P. Ben Ishai, A. J. Agranat, and Yu. Feldman, “Percolation of polar nanoregions: a dynamic approach to the ferroelectric phase transition,” J. Non-Cryst. Solids 353(47-51), 4422–4427 (2007).
[CrossRef]

2006 (1)

2004 (1)

P. Ishai, C. de Oliveira, Y. Ryabov, Y. Feldman, and A. Agranat, “Glass forming liquid kinetics manifested in a KTN:Cu crystal,” Phys. Rev. B 70(13), 132104 (2004).
[CrossRef]

2003 (1)

A. J. Agranat, “Optical lambda-switching at telecom wavelengths based on electroholography,” Top. Appl. Phys. 86, 133–161 (2003).
[CrossRef]

2002 (1)

G. Bitton, Yu. Feldman, and A. J. Agranat, “Relaxation processes of off-center impurities in KTN:Li crystals,” J. Non-Cryst. Solids 305(1-3), 362–367 (2002).
[CrossRef]

2001 (1)

Y. Girshberg and Y. Yacoby, “Off-centre displacements and ferroelectric phase transition in dilute KTa1−xNbxO3,” J. Phys. Condens. Matter 13(39), 8817–8830 (2001).
[CrossRef]

1999 (1)

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83(10), 1954–1957 (1999).
[CrossRef]

1998 (1)

G. Bitton, M. Razvag, and A. J. Agranat, “Formation of metastable ferroelectric clusters in K1-xLixTa1-yNbyO3:Cu,V at the paraelectric phase,” Phys. Rev. B 58(9), 5282–5286 (1998).
[CrossRef]

1997 (2)

A. J. Agranat, M. Razvag, M. Balberg, and V. Leyva, “Dipolar holographic gratings induced by the photorefractive process in potassium lithium tantalate niobate at the paraelectric phase,” J. Opt. Soc. Am. B 14(8), 2043–2048 (1997).
[CrossRef]

A. J. Agranat, M. Razvag, M. Balberg, and G. Bitton, “Holographic gratings by spatial modulation of the Curie-Weiss temperature in photorefractive K1-xLixTa1-yNbyO3:Cu,V,” Phys. Rev. B 55(19), 12818–12821 (1997).
[CrossRef]

1993 (1)

R. Hofmeister, A. Yariv, and A. Agranat, “Growth and characterization of the perovskite K1-yLiyTa1-xNbxO3:Cu,” J. Cryst. Growth 131(3-4), 486–494 (1993).
[CrossRef]

Agranat, A.

P. Ishai, C. de Oliveira, Y. Ryabov, Y. Feldman, and A. Agranat, “Glass forming liquid kinetics manifested in a KTN:Cu crystal,” Phys. Rev. B 70(13), 132104 (2004).
[CrossRef]

R. Hofmeister, A. Yariv, and A. Agranat, “Growth and characterization of the perovskite K1-yLiyTa1-xNbxO3:Cu,” J. Cryst. Growth 131(3-4), 486–494 (1993).
[CrossRef]

Agranat, A. J.

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nano-disordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011).
[CrossRef]

S. E. Lerner, P. Ben Ishai, A. J. Agranat, and Yu. Feldman, “Percolation of polar nanoregions: a dynamic approach to the ferroelectric phase transition,” J. Non-Cryst. Solids 353(47-51), 4422–4427 (2007).
[CrossRef]

A. Bitman, N. Sapiens, L. Secundo, A. J. Agranat, G. Bartal, and M. Segev, “Electroholographic tunable volume grating in the g44 configuration,” Opt. Lett. 31(19), 2849–2851 (2006).
[CrossRef] [PubMed]

A. J. Agranat, “Optical lambda-switching at telecom wavelengths based on electroholography,” Top. Appl. Phys. 86, 133–161 (2003).
[CrossRef]

G. Bitton, Yu. Feldman, and A. J. Agranat, “Relaxation processes of off-center impurities in KTN:Li crystals,” J. Non-Cryst. Solids 305(1-3), 362–367 (2002).
[CrossRef]

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83(10), 1954–1957 (1999).
[CrossRef]

G. Bitton, M. Razvag, and A. J. Agranat, “Formation of metastable ferroelectric clusters in K1-xLixTa1-yNbyO3:Cu,V at the paraelectric phase,” Phys. Rev. B 58(9), 5282–5286 (1998).
[CrossRef]

A. J. Agranat, M. Razvag, M. Balberg, and G. Bitton, “Holographic gratings by spatial modulation of the Curie-Weiss temperature in photorefractive K1-xLixTa1-yNbyO3:Cu,V,” Phys. Rev. B 55(19), 12818–12821 (1997).
[CrossRef]

A. J. Agranat, M. Razvag, M. Balberg, and V. Leyva, “Dipolar holographic gratings induced by the photorefractive process in potassium lithium tantalate niobate at the paraelectric phase,” J. Opt. Soc. Am. B 14(8), 2043–2048 (1997).
[CrossRef]

Balberg, M.

A. J. Agranat, M. Razvag, M. Balberg, and V. Leyva, “Dipolar holographic gratings induced by the photorefractive process in potassium lithium tantalate niobate at the paraelectric phase,” J. Opt. Soc. Am. B 14(8), 2043–2048 (1997).
[CrossRef]

A. J. Agranat, M. Razvag, M. Balberg, and G. Bitton, “Holographic gratings by spatial modulation of the Curie-Weiss temperature in photorefractive K1-xLixTa1-yNbyO3:Cu,V,” Phys. Rev. B 55(19), 12818–12821 (1997).
[CrossRef]

Bartal, G.

Ben Ishai, P.

S. E. Lerner, P. Ben Ishai, A. J. Agranat, and Yu. Feldman, “Percolation of polar nanoregions: a dynamic approach to the ferroelectric phase transition,” J. Non-Cryst. Solids 353(47-51), 4422–4427 (2007).
[CrossRef]

Bitman, A.

Bitton, G.

G. Bitton, Yu. Feldman, and A. J. Agranat, “Relaxation processes of off-center impurities in KTN:Li crystals,” J. Non-Cryst. Solids 305(1-3), 362–367 (2002).
[CrossRef]

G. Bitton, M. Razvag, and A. J. Agranat, “Formation of metastable ferroelectric clusters in K1-xLixTa1-yNbyO3:Cu,V at the paraelectric phase,” Phys. Rev. B 58(9), 5282–5286 (1998).
[CrossRef]

A. J. Agranat, M. Razvag, M. Balberg, and G. Bitton, “Holographic gratings by spatial modulation of the Curie-Weiss temperature in photorefractive K1-xLixTa1-yNbyO3:Cu,V,” Phys. Rev. B 55(19), 12818–12821 (1997).
[CrossRef]

Conti, C.

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nano-disordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011).
[CrossRef]

de Oliveira, C.

P. Ishai, C. de Oliveira, Y. Ryabov, Y. Feldman, and A. Agranat, “Glass forming liquid kinetics manifested in a KTN:Cu crystal,” Phys. Rev. B 70(13), 132104 (2004).
[CrossRef]

Della Pergola, R.

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83(10), 1954–1957 (1999).
[CrossRef]

DelRe, E.

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nano-disordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011).
[CrossRef]

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83(10), 1954–1957 (1999).
[CrossRef]

Feldman, Y.

P. Ishai, C. de Oliveira, Y. Ryabov, Y. Feldman, and A. Agranat, “Glass forming liquid kinetics manifested in a KTN:Cu crystal,” Phys. Rev. B 70(13), 132104 (2004).
[CrossRef]

Feldman, Yu.

S. E. Lerner, P. Ben Ishai, A. J. Agranat, and Yu. Feldman, “Percolation of polar nanoregions: a dynamic approach to the ferroelectric phase transition,” J. Non-Cryst. Solids 353(47-51), 4422–4427 (2007).
[CrossRef]

G. Bitton, Yu. Feldman, and A. J. Agranat, “Relaxation processes of off-center impurities in KTN:Li crystals,” J. Non-Cryst. Solids 305(1-3), 362–367 (2002).
[CrossRef]

Girshberg, Y.

Y. Girshberg and Y. Yacoby, “Off-centre displacements and ferroelectric phase transition in dilute KTa1−xNbxO3,” J. Phys. Condens. Matter 13(39), 8817–8830 (2001).
[CrossRef]

Hofmeister, R.

R. Hofmeister, A. Yariv, and A. Agranat, “Growth and characterization of the perovskite K1-yLiyTa1-xNbxO3:Cu,” J. Cryst. Growth 131(3-4), 486–494 (1993).
[CrossRef]

Ishai, P.

P. Ishai, C. de Oliveira, Y. Ryabov, Y. Feldman, and A. Agranat, “Glass forming liquid kinetics manifested in a KTN:Cu crystal,” Phys. Rev. B 70(13), 132104 (2004).
[CrossRef]

Lerner, S. E.

S. E. Lerner, P. Ben Ishai, A. J. Agranat, and Yu. Feldman, “Percolation of polar nanoregions: a dynamic approach to the ferroelectric phase transition,” J. Non-Cryst. Solids 353(47-51), 4422–4427 (2007).
[CrossRef]

Leyva, V.

Razvag, M.

G. Bitton, M. Razvag, and A. J. Agranat, “Formation of metastable ferroelectric clusters in K1-xLixTa1-yNbyO3:Cu,V at the paraelectric phase,” Phys. Rev. B 58(9), 5282–5286 (1998).
[CrossRef]

A. J. Agranat, M. Razvag, M. Balberg, and G. Bitton, “Holographic gratings by spatial modulation of the Curie-Weiss temperature in photorefractive K1-xLixTa1-yNbyO3:Cu,V,” Phys. Rev. B 55(19), 12818–12821 (1997).
[CrossRef]

A. J. Agranat, M. Razvag, M. Balberg, and V. Leyva, “Dipolar holographic gratings induced by the photorefractive process in potassium lithium tantalate niobate at the paraelectric phase,” J. Opt. Soc. Am. B 14(8), 2043–2048 (1997).
[CrossRef]

Ryabov, Y.

P. Ishai, C. de Oliveira, Y. Ryabov, Y. Feldman, and A. Agranat, “Glass forming liquid kinetics manifested in a KTN:Cu crystal,” Phys. Rev. B 70(13), 132104 (2004).
[CrossRef]

Sapiens, N.

Secundo, L.

Segev, M.

A. Bitman, N. Sapiens, L. Secundo, A. J. Agranat, G. Bartal, and M. Segev, “Electroholographic tunable volume grating in the g44 configuration,” Opt. Lett. 31(19), 2849–2851 (2006).
[CrossRef] [PubMed]

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83(10), 1954–1957 (1999).
[CrossRef]

Spinozzi, E.

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nano-disordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011).
[CrossRef]

Tamburrini, M.

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83(10), 1954–1957 (1999).
[CrossRef]

Toulouse, J.

J. Toulouse, “The three characteristic temperatures of relaxor dynamics and their meaning,” Ferroelectrics 369(1), 203–213 (2008).
[CrossRef]

Yacoby, Y.

Y. Girshberg and Y. Yacoby, “Off-centre displacements and ferroelectric phase transition in dilute KTa1−xNbxO3,” J. Phys. Condens. Matter 13(39), 8817–8830 (2001).
[CrossRef]

Yariv, A.

R. Hofmeister, A. Yariv, and A. Agranat, “Growth and characterization of the perovskite K1-yLiyTa1-xNbxO3:Cu,” J. Cryst. Growth 131(3-4), 486–494 (1993).
[CrossRef]

Ferroelectrics (1)

J. Toulouse, “The three characteristic temperatures of relaxor dynamics and their meaning,” Ferroelectrics 369(1), 203–213 (2008).
[CrossRef]

J. Cryst. Growth (1)

R. Hofmeister, A. Yariv, and A. Agranat, “Growth and characterization of the perovskite K1-yLiyTa1-xNbxO3:Cu,” J. Cryst. Growth 131(3-4), 486–494 (1993).
[CrossRef]

J. Non-Cryst. Solids (2)

S. E. Lerner, P. Ben Ishai, A. J. Agranat, and Yu. Feldman, “Percolation of polar nanoregions: a dynamic approach to the ferroelectric phase transition,” J. Non-Cryst. Solids 353(47-51), 4422–4427 (2007).
[CrossRef]

G. Bitton, Yu. Feldman, and A. J. Agranat, “Relaxation processes of off-center impurities in KTN:Li crystals,” J. Non-Cryst. Solids 305(1-3), 362–367 (2002).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. Condens. Matter (1)

Y. Girshberg and Y. Yacoby, “Off-centre displacements and ferroelectric phase transition in dilute KTa1−xNbxO3,” J. Phys. Condens. Matter 13(39), 8817–8830 (2001).
[CrossRef]

Nat. Photonics (1)

E. DelRe, E. Spinozzi, A. J. Agranat, and C. Conti, “Scale-free optics and diffractionless waves in nano-disordered ferroelectrics,” Nat. Photonics 5(1), 39–42 (2011).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (3)

A. J. Agranat, M. Razvag, M. Balberg, and G. Bitton, “Holographic gratings by spatial modulation of the Curie-Weiss temperature in photorefractive K1-xLixTa1-yNbyO3:Cu,V,” Phys. Rev. B 55(19), 12818–12821 (1997).
[CrossRef]

G. Bitton, M. Razvag, and A. J. Agranat, “Formation of metastable ferroelectric clusters in K1-xLixTa1-yNbyO3:Cu,V at the paraelectric phase,” Phys. Rev. B 58(9), 5282–5286 (1998).
[CrossRef]

P. Ishai, C. de Oliveira, Y. Ryabov, Y. Feldman, and A. Agranat, “Glass forming liquid kinetics manifested in a KTN:Cu crystal,” Phys. Rev. B 70(13), 132104 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

E. DelRe, M. Tamburrini, M. Segev, R. Della Pergola, and A. J. Agranat, “Spontaneous self-trapping of optical beams in metastable paraelectric crystals,” Phys. Rev. Lett. 83(10), 1954–1957 (1999).
[CrossRef]

Top. Appl. Phys. (1)

A. J. Agranat, “Optical lambda-switching at telecom wavelengths based on electroholography,” Top. Appl. Phys. 86, 133–161 (2003).
[CrossRef]

Other (3)

R. Blinc and B. Zeks, Soft Modes in Ferroelectrics and Antiferroelectrics (Elsevier, 1974).

A. Yariv and P. Yeh, Optical Waves in Crystals (John Wiley & Sons, 1984), Chapter 7.3.1.

M. E. Lines and A. M. Glass, Principles and Applications of Ferroelectrics and Related Materials (Clarendon, 1977), Chapter 4.

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Figures (7)

Fig. 1
Fig. 1

Schematics of crossed-polarizer setup.

Fig. 2
Fig. 2

Experimental CPS output for a) KTN at Tc + 4.0°C and b) KLTN at Tc + 4.4°C.

Fig. 3
Fig. 3

qualitative picture of the mean-field approximation Gibbs free energy in centrosymmetric crystal as function of polarization without field applied (left) and when electric field applied (right) with the parameters of the double well model used to derive the probability χ (for the definition of χ see the paragraph below).

Fig. 4
Fig. 4

A modulator setup with a crystal in a two phase mixture state.

Fig. 5
Fig. 5

Birefringence measurements in modulator configuration (right) with respective dielectric constants (left) of pure KTN 4°C above Tc (top) and KLTN 4.4°C above Tc (bottom) with Eq. (10) fitted to the birefringence measurement (solid red) and Eq. (19) fitted to the dielectric constant (dashed red). The experimental data is represented by solid black lines.

Fig. 6
Fig. 6

Determination of the dielectric constant parameters on the example of KLTN by fitting Eq. (19) to the experimental data collected at 4.4°C above Tc.

Fig. 7
Fig. 7

Determination of the birefringence parameters on the example of KLTN by fitting Eq. (10) to the experimental data collected at 4.4°C above Tc.

Tables (3)

Tables Icon

Table 1 Sample Properties

Tables Icon

Table 2 Fitting Parameters of the Dielectric Constant Part of the Model

Tables Icon

Table 3 Other Fitting Parameters—The Birefringence Part of the Model

Equations (23)

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x 1 2 n 11 2 + x 2 2 n 22 2 + x 3 2 n 33 2 = 1 ,
Δ ( 1 n i j 2 ) = k , l = 1 3 g i j k l P k P l ,
P k = ε E k ,
I = I 0 sin 2 ( Δ Φ 2 ) ,
Δ Φ = π n 0 3 ( g 11 g 12 ) ε 2 E 2 L λ ,
P ( n ) = 1 ( 2 π σ 2 ) 1 2 exp ( ( n < n > ) 2 2 σ 2 ) ,
P ( n ) = 1 ( 2 π L Δ L χ ( 1 χ ) ) 1 2 exp ( ( n L Δ L χ ) 2 2 L Δ L χ ( 1 χ ) ) .
Δ Φ | n = 2 π Δ n e f f L λ = 2 π Δ n f n Δ L + Δ n p ( N n ) Δ L λ = 2 π Δ L λ ( Δ n f n + Δ n p ( L Δ L n ) ) = 2 π Δ L λ ( ( Δ n f Δ n p ) n + Δ n p L Δ L ) = 2 π L λ Δ n p + 2 π Δ L n λ ( Δ n f Δ n p ) ,
I o u t = I i n n P ( n ) sin 2 ( Δ Φ 2 | n ) = = I i n ( 1 ( 2 π L Δ L χ ( 1 χ ) ) 1 2 exp ( ( n L Δ L χ ) 2 2 L Δ L χ ( 1 χ ) ) ) sin 2 ( π L λ ( Δ n p + Δ L n L ( Δ n f Δ n p ) ) ) d n ,
I o u t = I i n 1 2 [ 1 exp ( 8 π 2 Δ L L λ 2 ( Δ n f Δ n p ) 2 χ ( 1 χ ) ) cos [ 2 π L λ ( Δ n p + χ ( Δ n f Δ n p ) ) ] ] .
τ P = τ 0 exp ( V / k B T )
τ F = τ 0 exp ( [ V + Δ ] / k B T ) ,
d N F d t = N F τ 0 exp ( V / k B T ) + N P τ 0 exp ( [ V + Δ ] / k B T )
d N P d t = + N F τ 0 exp ( V / k B T ) N P τ 0 exp ( [ V + Δ ] / k B T ) ,
N F exp ( V / k B T ) N P exp ( [ V + Δ ] / k B T ) = 0 ,
χ = 1 1 + exp ( Δ / k B T ) ,
χ ( E ) = 1 1 + exp [ E E 0 χ σ ] .
Δ n P ( E ) = ( 1 2 ) n 0 3 g P ( P P ( E ) ) 2
Δ n F ( E ) = ( 1 2 ) n 0 3 [ g F ( P F ( E ) ) 2 + r F P F ( E ) + C 0 ] ,
P P , F ( E ) = 0 E ε s s P , F ( E ' ) d E ' ,
ε s s ( E ) = ε s s P [ 1 χ ( E ) ] + ε s s F χ ( E ) .
ε s s P = a + b s 2 + ( E E 0 P ) 2
ε s s F = c ,

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